Convergence analysis of an Inexact Infeasible Interior Point method for Semidefinite Programming

In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima,Megiddo and Mizuno (A primal-dual infeasible-interior-point algorithm for Linear Programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions; i.e., the linear systems defining the search directions can be solved with an accuracy that increases as the solution is approached. A convergence analysis is carried out and the global convergence of the method is prove

Variational density matrix optimization using semidefinite programming

We discuss how semidefinite programming can be used to determine the second-order density matrix directly through a variational optimization. We show how the problem of characterizing a physical or N -representable density matrix leads to matrix-positivity constraints on the density matrix. We then formulate this in a standard semidefinite programming form, after which two interior point methods are discussed to solve the SDP. As an example we show the results of an application of the method on the isoelectronic series of Beryllium.Comment: corrected typos, added do

Domain Decomposition for Stochastic Optimal Control

This work proposes a method for solving linear stochastic optimal control (SOC) problems using sum of squares and semidefinite programming. Previous work had used polynomial optimization to approximate the value function, requiring a high polynomial degree to capture local phenomena. To improve the scalability of the method to problems of interest, a domain decomposition scheme is presented. By using local approximations, lower degree polynomials become sufficient, and both local and global properties of the value function are captured. The domain of the problem is split into a non-overlapping partition, with added constraints ensuring $C^1$ continuity. The Alternating Direction Method of Multipliers (ADMM) is used to optimize over each domain in parallel and ensure convergence on the boundaries of the partitions. This results in improved conditioning of the problem and allows for much larger and more complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201

A sequential semidefinite programming method and an application in passive reduced-order modeling

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. In particular, a suitable symmetrization procedure needs to be chosen for the linearization of the complementarity condition. The choice of the symmetrization procedure can be shifted in a very natural way to certain linear semidefinite subproblems, and can thus be reduced to a well-studied problem. The resulting sequential semidefinite programming (SSP) method is a generalization of the well-known SQP method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class

Bounding stationary averages of polynomial diffusions via semidefinite programming

We introduce an algorithm based on semidefinite programming that yields increasing (resp. decreasing) sequences of lower (resp. upper) bounds on polynomial stationary averages of diffusions with polynomial drift vector and diffusion coefficients. The bounds are obtained by optimising an objective, determined by the stationary average of interest, over the set of real vectors defined by certain linear equalities and semidefinite inequalities which are satisfied by the moments of any stationary measure of the diffusion. We exemplify the use of the approach through several applications: a Bayesian inference problem; the computation of Lyapunov exponents of linear ordinary differential equations perturbed by multiplicative white noise; and a reliability problem from structural mechanics. Additionally, we prove that the bounds converge to the infimum and supremum of the set of stationary averages for certain SDEs associated with the computation of the Lyapunov exponents, and we provide numerical evidence of convergence in more general settings

A Quantum Interior Point Method for LPs and SDPs

We present a quantum interior point method with worst case running time $\widetilde{O}(\frac{n^{2.5}}{\xi^{2}} \mu \kappa^3 \log (1/\epsilon))$ for SDPs and $\widetilde{O}(\frac{n^{1.5}}{\xi^{2}} \mu \kappa^3 \log (1/\epsilon))$ for LPs, where the output of our algorithm is a pair of matrices $(S,Y)$ that are $\epsilon$-optimal $\xi$-approximate SDP solutions. The factor $\mu$ is at most $\sqrt{2}n$ for SDPs and $\sqrt{2n}$ for LP's, and $\kappa$ is an upper bound on the condition number of the intermediate solution matrices. For the case where the intermediate matrices for the interior point method are well conditioned, our method provides a polynomial speedup over the best known classical SDP solvers and interior point based LP solvers, which have a worst case running time of $O(n^{6})$ and $O(n^{3.5})$ respectively. Our results build upon recently developed techniques for quantum linear algebra and pave the way for the development of quantum algorithms for a variety of applications in optimization and machine learning.Comment: 32 page